Galois Groups and Genera of a kind of Quasi-cyclotomic Function Fields

TL;DR

This paper investigates the structure of Galois groups and genus formulas of a specific class of non-abelian, Galois extensions called quasi-cyclotomic function fields over rational function fields.

Contribution

It provides a detailed determination of the Galois group structures and explicit genus formulas for quasi-cyclotomic function fields, expanding understanding of their algebraic properties.

Findings

Galois groups are explicitly characterized

Genus formulas for quasi-cyclotomic fields are derived

Extension structures are classified

Abstract

We call a (q-1)-th Kummer extension of a cyclotomic function field a quasi-cyclotomic function field if it is Galois, but non-abelian, over the rational function field with the constant field of q elements. In this paper, we determine the structure of the Galois groups of a kind of quasi-cyclotomic function fields over the base field. We also give the genus formulae of them.